I feel you (and apparently some Youtubers) missed the point of the episode. It's not about "alcoholism is not a disease", hell just two seasons ago the series brought up the fact that Randy is an alcoholic (though in denial) and the episode brings to life that his family does have genetic predisposition for addiction of varying kinds (Randy's father is a compulsive gambler, Randy is an alcoholic, and Stan has addiction to FTP games) so so they're not saying he doesn't have a problem. Hell, Randy's actual alcoholism didn't really become a thing until a few seasons after this episode.

The real issue is that Randy is also a bonafide hypochondriac and telling him that his alcoholism is a disease made him worse (Stan pretty much says as much to the AA members) and the real issue of the story is that alcoholism is not something that can be treated on relying on a "higher power" to do so when even religious folks would agree that the individual themselves has to put some effort into reforming but Randy literally falls apart to the somewhat cult-ish practices of some AA groups who often use religion as a tool to coerce victims of alcoholism to seek the wrong kind of treatment as opposed to real medical help.

I saw the episode more as a "take that" to people using religion as the only means of getting past real world problems and preying on people who may not actually have a problem but have other insecurities that make them believe they have the issue. You know, like how pharmaceutical companies will sometimes use vague symptoms in their drug commercials to convince gullible people to believe that have some rare disease that only their drug can help. Kind of like that.

Regardless, South Park largely works on shock value, and while they do occasionally nail some good cultural commentary (largely by taking the piss out of it) it's still an entertainment show and not a series with focus on social enlightenment. It's all for laughs, and shouldn't be taken seriously.